What Is A Derivative Filter

Image derivatives can be computed by using small convolution filters of size 2 × 2 or 3 × three, such as the Laplacian, Sobel, Roberts and Prewitt operators.^[1] Still, a larger mask will generally give a meliorate approximation of the derivative and examples of such filters are Gaussian derivatives^[2] and Gabor filters.^[3] Sometimes high frequency noise needs to exist removed and this tin can be incorporated in the filter so that the Gaussian kernel will act as a band laissez passer filter.^[4] The use of Gabor filters^[5] in image processing has been motivated by some of its similarities to the perception in the human visual organisation.^[6]

The pixel value is computed every bit a convolution

${\displaystyle p'_{u}=\mathbf {d} \ast Thousand}$

where ${\displaystyle \mathbf {d} }$ is the derivative kernel and ${\displaystyle Thou}$ is the pixel values in a region of the image and ${\displaystyle \ast }$ is the operator that performs the convolution.

Sobel derivatives [edit]

The derivative kernels, known every bit the Sobel operator are defined as follows, for the ${\displaystyle u}$ and ${\displaystyle v}$ directions respectively:

${\displaystyle p'_{u}={\begin{bmatrix}+one&+2&+ane\\0&0&0\\-1&-2&-one\stop{bmatrix}}*\mathbf {Grand} \quad {\mbox{and}}\quad p'_{five}={\begin{bmatrix}+1&0&-1\\+2&0&-2\\+one&0&-one\stop{bmatrix}}*\mathbf {M} }$

where ${\displaystyle *}$ here denotes the 2-dimensional convolution operation.

This operator is separable and can be decomposed as the products of an interpolation and a differentiation kernel, so that, ${\displaystyle p'_{v}}$ , for an example can exist written every bit

${\displaystyle {\begin{bmatrix}+1&0&-1\\+2&0&-2\\+1&0&-1\end{bmatrix}}={\brainstorm{bmatrix}1\\two\\1\finish{bmatrix}}{\begin{bmatrix}+ane&0&-ane\end{bmatrix}}}$

Farid and Simoncelli derivatives [edit]

Farid and Simoncelli^{[vii] [eight]} propose to utilize a pair of kernels, one for interpolation and another for differentiation (compare to Sobel to a higher place). These kernels, of fixed sizes 5 x 5 and 7 x seven, are optimized so that the Fourier transform approximates their correct derivative human relationship.

In Matlab code the so called 5-tap filter is

                        k                                    =                                    [            0.030320                                    0.249724                                    0.439911                                    0.249724                                    0.030320            ];                        d                                    =                                    [            0.104550                                    0.292315                                    0.000000                                    -            0.292315                                    -            0.104550            ];                        d2                                    =                                    [            0.232905                                    0.002668                                    -            0.471147                                    0.002668                                    0.232905            ];                      

And the 7-tap filter is

                        grand                                    =                                    [                                    0.004711                                    0.069321                                    0.245410                                    0.361117                                    0.245410                                    0.069321                                    0.004711            ];                        d                                    =                                    [                                    0.018708                                    0.125376                                    0.193091                                    0.000000                                    -            0.193091                                    -            0.125376                                    -            0.018708            ];                        d2                                    =                                    [                                    0.055336                                    0.137778                                    -            0.056554                                    -            0.273118                                    -            0.056554                                    0.137778                                    0.055336            ];                      

As an instance the start lodge derivatives can be computed in the following using Matlab in club to perform the convolution

                                                Iu                                    =                                    conv2            (            d            ,                                    grand            ,                                    im            ,                                    'same'            );                                    % derivative vertically (wrt Y)                        4                                    =                                    conv2            (            k            ,                                    d            ,                                    im            ,                                    'aforementioned'            );                                    % derivative horizontally (wrt X)                      

It is noted that Farid and Simoncelli have derived showtime derivative coefficients which are more accurate compared to the ones provided higher up. However, the latter are consequent with the 2nd derivative interpolator and, therefore, are improve to employ if both the starting time and second derivatives are sought. In the opposite case, when only the first derivative is desired, the optimal get-go derivative coefficients should be employed; more details can be plant in their paper.

Hast derivatives [edit]

Derivative filters based on arbitrary cubic splines was presented by Hast.^[9] He showed how both first and 2d order derivatives can exist computed more than correctly using cubic or trigonometric splines. Efficient derivative filters need to exist of odd length so that the derivative is computed for the central pixel. However, any cubic filter is fitted over iv sample points, giving a centre that falls between pixels. This is solved by a double filtering approach giving filters of size 7 x 7. The idea is to first filter by interpolation and then that the interpolated value between pixels are obtained, whereafter the procedure is repeated using a derivative filters, where the centre value now falls on pixel centres. This can easily be proved past the associative law for convolution

${\displaystyle p'_{u}=\mathbf {d} \ast (\mathbf {m} \ast G)=(\mathbf {d} \ast \mathbf {yard} )\ast Yard}$

Therefore the convolution kernel for computing the derivative ${\displaystyle \mathbf {k_{d}} }$ using an interpolating kernel ${\displaystyle \mathbf {k} }$ and a derivative kernel ${\displaystyle \mathbf {d} }$ becomes

${\displaystyle \mathbf {k_{d}} =\mathbf {d} \ast \mathbf {thousand} }$

Too keep in mind that convolution is commutative, so that the guild of the two kernels does not matter and it is besides possible to insert a second order derivative every bit well as a first order derivative kernel. These kernels are derived from the fact that whatever spline surface tin be fitted over a square pixel region, compare to Bezier surfaces. Hast proves that such a surface tin can be performed as a separable convolution

${\displaystyle p(u,five)=\mathbf {u} ^{T}MGM^{T}\mathbf {5} =M^{T}\mathbf {u} \otimes \mathbf {v} ^{T}M\ast G}$

where ${\displaystyle M}$ is the spline basis matrix, ${\displaystyle \mathbf {u} }$ and ${\displaystyle \mathbf {v} }$ are vectors containing the variables ${\displaystyle u}$ and ${\displaystyle five}$ , such equally

${\displaystyle \mathbf {u} =[u^{3},u^{2},u,1]^{T}}$

${\displaystyle \mathbf {v} =[v^{3},v^{2},v,1]^{T}}$

The convolution kernels can now be set to

${\displaystyle \mathbf {k} =\mathbf {u} ^{T}M=(M^{T}\mathbf {5} )^{T}}$

${\displaystyle \mathbf {d} ={\frac {\fractional \mathbf {u} ^{T}}{\fractional u}}M=\left(M^{T}{\frac {\fractional \mathbf {v} }{\partial v}}\right)^{T}}$

${\displaystyle \mathbf {d^{2}} ={\frac {\partial ^{2}\mathbf {u} ^{T}}{\fractional u^{two}}}M=\left(M^{T}{\frac {\partial ^{2}\mathbf {v} }{\partial v^{2}}}\right)^{T}}$

The first order derivatives at the central pixel are hence computed as

${\displaystyle D_{u}={\frac {\partial \mathbf {u} ^{T}}{\partial u}}Thou\ast \mathbf {u} ^{T}MGM^{T}\mathbf {5} \ast M^{T}\mathbf {5} =\mathbf {d} \ast \mathbf {k} \otimes (\mathbf {k} \ast \mathbf {k} )^{T}\ast K}$

and

${\displaystyle D_{v}=\mathbf {u} ^{T}M\ast \mathbf {u} ^{T}MGM^{T}\mathbf {v} \ast Thou^{T}{\frac {\partial \mathbf {5} }{\partial v}}=\mathbf {one thousand} \ast \mathbf {k} \otimes (\mathbf {d} \ast \mathbf {1000} )^{T}\ast G}$

Likewise, with the second society derivative kernels are

${\displaystyle D_{u}^{2}={\frac {\fractional ^{2}\mathbf {u} ^{T}}{\fractional u^{2}}}M\ast \mathbf {u} ^{T}MGM^{T}\mathbf {v} \ast M^{T}\mathbf {v} =\mathbf {d^{2}} \ast \mathbf {yard} \otimes (\mathbf {k} \ast \mathbf {k} )^{T}\ast 1000}$

and

${\displaystyle D_{5}^{two}=\mathbf {u} ^{T}M\ast \mathbf {u} ^{T}MGM^{T}\mathbf {v} \ast M^{T}{\frac {\partial ^{2}\mathbf {five} }{\partial five^{2}}}=\mathbf {k} \ast \mathbf {thou} \otimes (\mathbf {d^{ii}} \ast \mathbf {k} )^{T}\ast K}$

The cubic spline filter is evaluated in its eye ${\displaystyle u=v=0.v}$ and therefore

${\displaystyle \mathbf {u} =\mathbf {v} =[(0.5)^{3},(0.v)^{2},0.v,i]^{T}=[0.125,0.25,0.5,1]^{T}}$

Likewise the first club derivatives becomes

${\displaystyle {\frac {\fractional \mathbf {u} (0.five)}{\partial u}}={\frac {\partial \mathbf {v} (0.5)}{\partial v}}=[three\cdot (0.five)^{2},two\cdot (0.5),ane,0]^{T}=[0.75,1,ane,0]^{T}}$

And in a similar manner the 2nd lodge derivatives are

${\displaystyle {\frac {d^{two}\mathbf {u} (0.5)}{u^{2}}}={\frac {d^{2}\mathbf {v} (0.5)}{dv^{2}}}=[6\cdot (0.5),ii,0,0]^{T}=[iii,2,0,0]^{T}}$

Any cubic filter can be applied and used for calculating the image derivates using the above equations, such as Bézier, Hermite or B-splines.

The example in below in Matlab use the Catmull-Rom spline to compute the derivatives

                                                Thou                                    =                                    [            1            ,            -            three            ,            3            ,            -            1            ;                                    -            1            ,            iv            ,            -            5            ,            two            ;                                    0            ,            1            ,            0            ,            -            1            ;                                    0            ,            0            ,            ii            ,            0            ]                                    *                                    0.5            ;                        u                                    =                                    [            0.125            ;            0.25            ;            0.5            ;            ane            ];                        upward                                    =                                    [            0.75            ;            i            ;            1            ;            0            ];                        d                                    =                                    upwardly            '*            G            ;                        k                                    =                                    u            '*            M            ;                        Iu                                    =                                    conv2            (            conv            (            d            ,            k            ),                                    conv            (            one thousand            ,            k            ),                                    im            ,            'aforementioned'            );                                    % vertical derivative (wrt Y)                        Iv                                    =                                    conv2            (            conv            (            m            ,            k            ),                                    conv            (            d            ,            k            ),                                    im            ,            'aforementioned'            );                                    % horizontal derivative (wrt X)                      

Other approaches [edit]

Steerable filters can be used for computing derivatives^[x] Moreover, Savitzky and Golay^[11] propose a to the lowest degree-squares polynomial smoothing approach, which could be used for computing derivatives and Luo et al^[12] discuss this arroyo in further detail. Scharr^{[13] [14] [15]} shows how to create derivative filters by minimizing the error in the Fourier domain and Jähne et al^[16] discuss in more than detail the principles of filter design, including derivative filters.

References [edit]

1. ^ Pratt, W.K., 2007. Digital epitome processing (fourth ed.). John Wiley & Sons, Inc. pp. 465–522
2. ^ H. Bouma, A. Vilanova, J.O. Bescós, B.M.T.H. Romeny, F.A. Gerritsen, Fast and accurate gaussian derivatives based on b-splines, in: Proceedings of the 1st International Conference on Scale Space and Variational Methods in Computer Vision, Springer-Verlag, Berlin, Heidelberg, 2007, pp. 406–417.
3. ^ P. Moreno, A. Bernardino, J. Santos-Victor, Improving the sift descriptor with smoothen derivative filters, Pattern Recognition Letters thirty (2009) 18–26.
4. ^ J.J. Koenderink, A.J. van Doorn, Generic neighborhood operators, IEEE Trans. Design Anal. Mach. Intell. 14 (1992) 597–605.
5. ^ D. Gabor, Theory of communication, J. Inst. Electr. Eng. 93 (1946) 429–457.
6. ^ J.G. Daugman, Consummate discrete 2-D Gabor transforms by neural networks for paradigm analysis and compression, IEEE Trans. Acoust. Speech Signal Procedure. 36 (1988) 1169–1179.
7. ^ H. Farid and E. P. Simoncelli, Differentiation of discrete multi-dimensional signals, IEEE Trans Image Processing, vol.thirteen(4), pp. 496--508, Apr 2004.
8. ^ H. Farid and Eastward. P. Simoncelli, Optimally Rotation-Equivariant Directional Derivative Kernels, Int'l Conf Computer Analysis of Images and Patterns, pp. 207--214, Sep 1997.
9. ^ A. Hast., "Simple filter design for first and 2nd order derivatives by a double filtering approach", Pattern Recognition Letters, Vol. 42, no.1 June, pp. 65--71. 2014.
10. ^ Westward.T. Freeman, E.H. Adelson, The design and use of steerable filters, IEEE Trans. Pattern Anal. Mach. Intell. thirteen (1991) 891–906.
11. ^ A. Savitzky, M.J.E. Golay, Smoothing and differentiation of data by simplified least squares procedures, Anal. Chem. 36 (1964) 1627–1639.
12. ^ J. Luo, K. Ying, P. He, J. Bai, Properties of savitzky–golay digital differentiators, Digit. Bespeak Procedure. xv (2005) 122–136.
13. ^ H. Scharr, Optimal second gild derivative filter families for transparent motion estimation, in: 1000. Domanski, R. Stasinski, M. Bartkowiak (Eds.), EUSIPCO 2007.
14. ^ Scharr, Hanno, 2000, Dissertation (in German), Optimal Operators in Digital Epitome Processing .
15. ^ B. Jähne, H. Scharr, and Due south. Körkel. Principles of filter design. In Handbook of Calculator Vision and Applications. Academic Press, 1999.
16. ^ B. Jähne, P. Geissler, H. Haussecker (Eds.), Handbook of Figurer Vision and Applications with Cdrom, 1st ed., Morgan Kaufmann Publishers Inc., San Francisco, CA, The states, 1999, pp. 125–151 (Chapter six).

External links [edit]

• derivative5.k Farid and Simoncelli: 5-Tap 1st and 2nd discrete derivatives.
• derivative7.m Farid and Simoncelli: vii-Tap 1st and 2nd discrete derivatives
• kernel.m Hast: 1st and 2nd discrete derivatives for Cubic splines, Catmull-Rom splines, Bezier splines, B-Splines and Trigonometric splines.

What Is A Derivative Filter,

Source: https://en.wikipedia.org/wiki/Image_derivative

Posted by: stricklandwhousen.blogspot.com

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